Classification of the congruence classes of \(A_n^5\) (\(n \geqslant 6\)) with 2-torsion free homology (Q2197855)

Let \(\mathbf{A}_n^k\) \((n\geq k+1)\) denote the homotopy category consisting of all \((n-1)\)-connected finite CW complexes \(X\) with \(\dim X\leq n+k\), and let \(\mathbf{F}_{n(2)}^k\) be the full subcategory of \(\mathbf{A}_n^k\) of polyhedra with \(2\)-torsion free homology groups. A polyhedron \(X\) is called indecomposable if there is a homotopy equivalence \(X\simeq X_1\vee X_2\), then this implies that one of \(X_k\) (\(k=1,2)\) is contractible. Two polyhedra \(X\) and \(Y\) are called congruent if there is a homotopy equivalence \(X\vee Z\simeq Y\vee Z\) for some polyhedron \(Z\). The authors study the homotopy classification problem of indecomposable homotopy types in \(\mathbf{A}_n^k\). In this paper they classify all indecomposable congruence classes of homotopy types in \(\mathbf{F}^5_{n(2)}\) explicitly. Their proof is based on the matrix problem technique established by \textit{H.-J. Baues} and \textit{Y. Drozd} [Expo. Math. 17, No. 2, 161--179 (1999; Zbl 0942.55010)] which was developed in the classification of representations of algebras.